MATH 10560: CALCULUS II TRIGONOMETRIC FORMULAS Basic Identities The functions cos(θ) and sin(θ) are defined to be the x and y coordinates of the point at an angle of θ on the unit circle. Therefore, sin(−θ) = − sin(θ), cos(−θ) = cos(θ), and sin2 (θ) + cos2 (θ) = 1. The other trigonometric functions are defined in terms of sine and cosine: tan(θ) = sin(θ)/ cos(θ) sec(θ) = 1/ cos(θ) cot(θ) = csc(θ) = cos(θ)/ sin(θ) = 1/ tan(θ) 1/ sin(θ) Dividing sin2 (θ) + cos2 (θ) = 1 by cos2 (θ) or sin2 (θ) gives tan2 (θ) + 1 = sec2 (θ) and 1 + cot2 (θ) = csc2 (θ). Addition Formulas The following two addition formulas are fundamental: sin(A + B) cos(A + B) = sin(A) cos(B) + cos(A) sin(B) = cos(A) cos(B) − sin(A) sin(B) They can be used to prove simple identities like sin(π/2 − θ) = sin(π/2) cos(θ) + cos(π/2) sin(θ) = cos(θ), or cos(x − π) = cos(x) cos(π) − sin(x) sin(π) = − cos(x). If we set A = B in the addition formulas we get the double-angle formulas: sin(2A) = 2 sin(A) cos(A) cos(2A) = cos2 (A) − sin2 (A) The formula for cos(2A) is often rewritten by replacing cos2 (A) with 1 − sin2 (A) or replacing sin2 (A) with 1 − cos2 (A) to get cos(2A) = 1 − 2 sin2 (A) cos(2A) = 2 cos2 (A) − 1 Solving for sin2 (A) and cos2 (A) yields identities important for integration: 1 1 sin2 (A) = (1 − cos(2A)) cos2 (A) = (1 + cos(2A)) 2 2 The addition formulas can also be combined to give other formulas important for integration: sin A sin B = cos A cos B = sin A cos B = 1 2 [cos(A − B) − cos(A + B)] 1 2 [cos(A − B) + cos(A + B)] 1 2 [sin(A − B) + sin(A + B)] Derivatives and Integrals sin′ (x) = cos(x) cos′ (x) = − sin(x) tan′ (x) = sec2 (x) R R sin(x) dx = − cos(x) + C R cos(x) dx = sin(x) + C tan(x) dx = ln | sec(x)| + C sec′ (x) = sec(x) tan(x) csc′ (x) = − csc(x) cot(x) cot′ (x) = − csc2 (x) R R sec(x) dx = ln | sec(x) + tan(x)| + C R csc(x) dx = ln | csc(x) − cot(x)| + C cot(x) dx = − ln | csc(x)| + C